Garvin Melles
Let V be the universe of sets and V_α the sets of rank \leqα. We develop some axiom schemata for set theory based on the following three assumptions: 1. V \models ZFC 2. V is large with respect to the class of ordinals 3. V is large with respect to each of the V_α
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We give arguments for and prove the consistency of some internal forcing axioms.
This paper is a technical continuation of ``Natural Axiom Schemata Extending ZFC. Truth in the Universe?'' In that paper we argue that $CIFS$ is a natural axiom schema for the universe of sets. In particular it is a natural closure condition on $V$ and a natural generalization of $IFS(L).$ Here we shall prove the consistency of $ZFC\ +\ CIFS$ relative to the existence of a transitive model of $ZFC$ using the compactness theorem together with a class forcing.
We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.
We characterize those standard models M of ZFC which are embeddable, as the class of all standard sets, in a model of internal set theory IST. The necessary and sufficient condition is that 1) there is a wellordering < of M which does not destroy the ZFC schemata, and 2) the truth relation for (M,<) does not destroy Separation. The result is interpreted as the answer for the question in the title.
We study models of HST, a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the language containing the membership and standardness predicates, and Saturation for well-orderable families of internal sets. This theory admits an adequate formulation of the isomorphism property IP: "any two elementarily equivalent internally presented structures of a wellorderable language are isomorphic." IP implies, for instance, that all infinite internal sets are equinumerous, and there exists a unique (modulo isomorphisms) internal elementary extension of th…
Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of $M$ is a transitive set of the form $M[{\bf G}]$ where ${\bf G}$ is an $M$-complete ultrafilter on ${\bf B}.$) We prove that 1. If ${\bf G}$ is a $^*$forcing complete ultrafilter on ${\bf B},$ then $M[{\bf G}]\models ZFC.$ 2. Let $H\sub M.$ If there is a least transitive model $N$ such that $H\in M,$ $Ord^M=Ord^N,$ and $N\models ZFC,$ then we denote $N$ by $…
A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is presented in which L=HOD $\subsetneqq$ L$[a]$=HNT $\subsetneqq$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe.
Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_δ$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_δ$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $δ$, using a ground model parameter. In particular, this means that models of…
In light of the celebrated theorem of Vopěnka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amena…
We prove it consistent relative to ZFC that all nontrivial forcings of size $\aleph _1$ add a Cohen real.
We consider the modality "$φ$ is true in every $σ$-centered forcing extension", denoted $\squareφ$, and its dual "$φ$ is true in some $σ$-centered forcing extension", denoted $\lozengeφ$ (where $φ$ is a statement in set theory), which give rise to the notion of a "principle of $σ$-centered forcing". We prove that if ZFC is consistent, then the modal logic of $σ$-centered forcing, i.e. the ZFC-provable principles of $σ$-centered forcing, is exactly $\mathsf{S4.2}$. We also generalize this result to other related classes of forcing.
We give characterizations for the (in ZFC unprovable) sentences ``Every $\Sb{1}{2}$--set is measurable" and ``Every $\Db{1}{2}$--set is measurable" for various notions of measurability derived from well--known forcing partial orderings.
Typically, set theorists reason about forcing constructions in the context of ZFC. We show that without AC, several simple properties of forcing posets fail to hold, one of which answers Miller's question from arXiv:0704.3998.
A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all s…
The effects of forcing on exacting and ultraexacting cardinals
本文研究集合论强制对exacting与ultraexacting基数的影响，证明I2嵌入产生具有真类exacting基数的ZFC模型，并探讨HOD猜想下相关性质及强制扩张。
Jensen $\varDelta^1_3$ reals by means of ZFC$^-$ or second order Peano arithmetic
本文解决 Jensen 构造在二阶 Peano 算术或 ZFC$^-$ 内重现的难题，通过在 forcing 构造中仅消灭可定义反链来规避 CCC 证明障碍，并定义了极小 $\varPi^1_2$ 单例模型。
Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.
A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.
We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies ZFC. In doing so, we remodularized Paulson's ZF-Constructibility library.
PFA and the definability of the nonstationary ideal
该研究在含超紧基数的ZFC模型基础上构造出满足PFA的模型，证明ω₁上的非平稳理想在H_{ℵ₂}中可由参数Π₁-定义，解决了PFA与非平稳理想可定义性的关联问题。
We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbfΔ^1_2$ maximal independent set. As a main application we get the consistency of $\mathfrak{r} = \mathfrak{u} = \mathfrak{i} = ω_2$ together with the existence of a $Δ^1_2$ ultrafilter, a $Π^1_1$ maximal independent family and a $Δ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.
In this paper we show how to build a model of $ZFC$ such that all its inner models satisfying the Axiom of Choice are well-ordered with respect to inclusion, and that said ordering is of arbitrary height (including possibly $Ord$ high). We do this by iterating $κ$-Sacks forcing for ever-increasing $κ$, while showing that such forcings do not add any unexpected intermediate inner models.
Forcing the $Π^1_3$-Reduction Property and a Failure of $Π^1_3$-Uniformization
本文通过在可构造宇宙中强制得到 $Π^1_3$-还原性质模型，将大型基数强度从 $M_1^{\#}$ 降低到 ZFC，并首次证明 $Π^1_3$-统一化原理在此模型中失效。
Forcing upper $Σ$-uniformization in the presence of lower $Π$-reduction or uniformization
该研究提出结合射影分层两侧力迫均匀化的方法，构造了满足特定均匀化与归约关系的集合论宇宙，并降低了相关均匀化命题的上界。
Terminal Absoluteness of Collapse Forcings
本文探讨了坍缩forcing在项目性通用绝对性中的必要性，证明在大基数假设下，坍缩forcing的绝对性等价于任意forcing的绝对性，并指出在低项目性层次下结果在ZFC中成立，揭示了坍缩forcing的终端性。
We show that the set of codes for Ramsey positive analytic sets is $\mathbfΣ^1_2$-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is $\mathbfΣ^1_1$-complete. This shows a close resemblance between the Sacks forcing and the Mathias forcing. In particular, we get that the $σ$-ideal of Ramsey null sets is not ZFC-correct. This solves a problem posed by Ikegami, Pawlikowski and Zapletal.
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Gr…
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Gr…
The formal verification of the ctm approach to forcing
本研究借助Isabelle证明助手与Isabelle/ZF库，形式化验证了ZFC可数传递模型的力迫兼纳扩张构造，并明确了替换公理模式中21个关键实例的作用，为连续统假设相关独立性结果提供了机器验证基础。
We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $ω_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{\aleph_0}=\aleph_2$ and all $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ collapse $\aleph_2,$ and hence are forcing equivalent to $Add(\aleph_1, 1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and…
Assume ZFC. Let $κ$ be a cardinal. A ${<κ}$-ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<κ}$. The $κ$-mantle is the intersection of all ${<κ}$-grounds. We prove that certain partial choice principles in the $κ$-mantle are the consequence of $κ$ being inaccessible/weakly compact, and some other related facts.
Dependent Choice, Properness, and Generic Absoluteness
本文证明依赖选择原理足以发展恰当力迫理论，并在大基数下保证Chang模型的泛型绝对性，从而确立ZF+DC作为经典数学和集合论重要部分的自然基础。
With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S …
$Σ_n$-correct Forcing Axioms
本文提出了扩展ZFC的$Σ_n$-correct力迫公理族，证明了不同层级公理与对应大基数的相容性，还研究了其力迫保持性、与其他公理的关系及数学推论。
A torsion-free abelian group B of arbitrary rank is called a B_1-group if Bext^1(B,T)=0 for every torsion abelian group T, where Bext^1 denotes the group of equivalence classes of all balanced exact extensions of T by B. It is a long-standing problem whether or not the class of B_1-groups coincides with the class of B_2-groups. A torsion-free abelian group B is called a B_2-group if there exists a continuous well-ordered ascending chain of pure subgroups, 0=B_0 subset B_1 subset ... subset B_alpha subset ... subset B_lambda =B=bigcup_{alpha in lambda} B_alpha such that B_{alpha+1}=B_alpha+G_al…
Twins: non-isomorphic models forced to be isomorphic. Part I
本文研究哪些完备理论存在非同构模型在力迫扩张后变得同构，给出具有独立性质的理论在力迫不添加新ω-序列时的充分条件，并重点讨论向树添加新分支的力迫偏序，属于分类纲领与Main Gap纲领的一部分。
Menger and consonant sets in the Sacks model
本文通过迭代Sacks力迫和拓扑博弈，证明Cantor立方体中基数为连续统的全不完美Menger集的存在性独立于ZFC，并分析了Sacks模型中Hurewicz与consonant子集的结构。
Local mantles of $L[x]$
本文在 Woodin 基数与真类多可测基数假设下，利用 Woodin 对 HOD 的分析方法研究 $L[x]$ 的局部 $κ$-mantle，证明其满足 ZFC+GCH 并含 Woodin 基数，且为完全可迭代策略 mouse。
Naturality and Definability III
本文探究范畴论自然性与模型论可定义性的关联，证明温和条件下自然性蕴含可定义性，构造出满足可一致化构造均为弱自然的ZFC传递模型，还证明公式表示的自然构造可无额外参数一致可定义，回答了Hodges与Shelah的相关问题。
Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions
开发迫使$\lim^1 \mathbf{A} \ne 0$的方法，扩展了之前关于$\lim^k \mathbf{A}$（$k \geq 2$）的非零结果，给出特定条件下相对ZFC一致的结论，还表明在许多有限支持迭代强迫扩张中$\lim^1 \mathbf{A} = 0$成立。
A foundation for deductive mathematics
本文指出了ZFC公理体系在标准数学实践中的不一致性，通过引入“相容极限公理”构建了与极大模型对应的集合论基础，证明主流实践隐含于该理论中，并指出过去百年的公理集合论与标准实践无关。
Forcing With Copies of Uncountable Ordinals
本文研究由不可数序数副本构成的偏序集，在ZFC下证明了其要么σ-闭且保持ω₁，要么完全嵌入特定代数并将ω₂坍缩为ω。
Ordinal definability in $L[\mathbb{E}]$
本文研究了tame mouse模型中V=HOD_x的成立条件，证明了扩展序列在某些索引上可定义，并探讨了HOD^M与premouse的关系，以及无proper grounds的性质。
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